Affine Moment Invariants of Vector Fields

Vector fields are a special kind of multidimensional data. In each pixel, the field is assigned to a vector that shows the direction and the magnitude of the quantity, which has been measured. To detect the patterns of interest in the field, special matching methods must be developed. A 2D vector field f(x) can be mathematically described as a pair of scalar fields f(x) = (f1(x,y),f2(x,y)). At each point x = (x,y), the value of f(x) shows the orientation and the magnitude of the measured vector. Scalar field fi(x) can be understood as a graylevel image.

The geometric transformations of the vector fields are slightly different from the transformations of the images. The total affine transformation without translationf → f' acts simultaneously in spatial and function domains

f'(x) = Bf ( A-1x),

where A and B are regular matrices and f is a vector field. If A ≠ B, the transformation is called independent total affine transformation of field f. Matrix A is called inner transformation matrix, while matrix B is called outer transformation matrix.

In the total affine transformation with translation, the inner part A-1x is replaced with A-1 (x−t). However, for pattern detection via template matching it is irrelevant to include the translation into the deformation model, because the shift is the key parameter we want to detect.

Sometimes, vector fields are transformed by a slightly simpler transformation in which A = B. Such a model is called special total affine transformation and captures one of the basic properties of vector fields - if the field is transformed in the space domain, the function domain (i.e. the vector values) are transformed by the same transformation. The scenarios where A ≠ B are rare but may happen as well if, for instance, the measuring device exhibits different calibrations for inner and outer part.

The invariants of the vector fields to the independent total affine transformation can be generated as

where Ckj=xkyj−xjyk and Fkj=f1(xk,yk)f2(xj,yj)−f1(xj,yj)f2(xk,yk). The independent invariants generated according to this formula by the graphs up to the 9 edges of the first type are in the attachment file "afinvectc9indep.pdf". They were selected from the irreducible invariants in the attachment file "afinvectc9s.pdf".

The invariants of the vector fields to the special total affine transformation can be generated as

where Dkj=yjf1(xk,yk)−xjf2(xk,yk). The independent invariants generated according to this formula by the graphs up to the 9 edges of all types are in the attachment file "afinvectts9indep.pdf". They were selected from the irreducible invariants in the attachment file "afinvectts9.pdf".

The detailed description of the graph generation for both types of invariants is in "affine_vectorfields_graph_generation.pdf".
The videos with results of the experiment with Kármán street can be seen here.